How a Simple One-Third Relationship Reveals the Average Value of Triangular Constructions
This article will introduce you to “Modified Triangular Sums”. More importantly I will refer to what I have named, the “One-Third Principle”.
While exploring modified triangular sums, I noticed an interesting geometric relationship quite by accident, while attempting to find a modified Triangular number. This relationship which I have named the One-Third Principle for lack of a better term, provides a shortcut to finding a tetrahedral number or any total sum from the same triangular format.
Just a few thoughts to take into account. I use a right angle triangle rather than the well known equilateral triangle for summation of terms. Second, there already exists mathematical expressions that can be used to find these total sums. But I feel these expressions ‘hide’ the relationship of the number values to the triangular geometry.
First, we will review Triangular numbers:
The Standard Triangular-number Construction
